Math Project-based Learning

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"What science can there be more noble, more excellent, more useful for men, more admirably high and demonstrative, than this of mathematics?" (Benjamin Franklin; American author, printer, scientist, inventor, and diplomat; 1706–1790.)


Summary

"Tell me, and I will forget. Show me, and I may remember. Involve me, and I will understand." (Confucius; Chinese thinker and social philosopher; 551-479 B.C.)

This document provides an introduction to uses of Project-based Learning (PBL) in math education. It includes arguments supporting use of PBL and it includes a number of examples that can be adapted for use in a wide range of math courses at the pre-college level and in teacher education.

Project-based learning is a way to involve students in learning and using math. Project-based learning is a good vehicle for helping students make progress on a number of math educational goals not directly covered in the "traditional" math curriculum. Not least among these goals is to help make math education a pleasant and rewarding discipline of study that adults will look back on with fond memories.

The target audience for this document is preservice and inservice K-12 teachers who teach math, and teachers of such teachers. Keep in mind that the overriding goal or purpose of this document is to improve math education. As a teacher, you should consider making use of project-based learning when:

  1. You believe it will help to improve the quality of the math education your students are getting; and
  2. You believe it will help you learn more about teaching math and the ways in which your students learn math.

Introduction to Project and Problem

"A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas." (G.H. Hardy; English mathematician; 1877–1947.)

The acronym PBL is used to represent both project-based learning and problem-based learning. The two topics overlap, but are not the same.

PBL Venn.jpeg

The main focus in this document is on project-based learning. There, a student or a team of students—perhaps even the whole class or whole school—works on a project over an extended period of time. The student or students have some choice (indeed, perhaps a great deal of choice) on the topic they will address. Project-based learning is oriented toward producing a product, performance, or presentation.

A math PBL lesson typically has broader goals than just learning math content that will be assessed via traditional math tests. For example, One standard goal in math PBL is to help increase student levels of Math Maturity. Quoting from the Math Maturity website:

It is well recognized that some rote memory learning is quite important in math education. However, most of this rote learning suffers from a lack of long-term retention, and from the learner’s inability to transfer this learning to new, challenging problem situations both within the discipline of math as well as to math-related problem situations outside the discipline of math.
Thus, math education has moved in the direction of placing much more emphasis on learning for understanding and for solving novel (non-routine) problems. There is substantial emphasis on learning some “big ideas” and gaining math-related "habits of mind and thinking skills" that will last a lifetime.
There is still more to achieving a useful level of math maturity. A student needs to learn how to learn math, how to self-assess his or her level of math content knowledge, skills, and math maturity, how to relearn math that has been forgotten or partially forgotten, how to make effective use of online sources of math information and instruction, and so on.

A Math Project Example

Here is an example of math project-based learning. Students are given a general project area, and they are asked to work on it individually or in teams:

Select an academic discipline other than math. Investigate roles of math in helping to represent and solve the problems in this discipline. Pay special attention to what areas of math are important in the academic discipline you select, and why they are important. The specific assignment is to produce a paper and do an oral presentation that can help inform your fellow students.

This type of assignment could well extend over a number of weeks, with much of the required work being done outside of class. If this assignment is given in a specific math class (such as geometry or algebra) then the assignment might include the requirement that the study focus on roles of the math being studied in the class and that these be emphasized in the project.

The paper and presentation might become part of a student's overall academic portfolio or math portfolio. As a teacher in this setting, you might want to emphasize the idea of students developing a paper that will be useful to other students in the class, students in other similar classes, and future students in the class. And, of course, the work done by your students helps to represent the quality of instruction you are providing for your students. The intended audience is much larger than just the teacher!

For some general information about project-based learning see:

A Math Problem Example

This section provides some information about problem-based learning and how it differs from project-based learning. Quoting from an ERIC Digest article on Problem-based Learning in Mathematics:

Problem-Based Learning (PBL) describes a learning environment where problems drive the learning. That is, learning begins with a problem to be solved, and the problem is posed in such a way that students need to gain new knowledge before they can solve the problem. Rather than seeking a single correct answer, students interpret the problem, gather needed information, identify possible solutions, evaluate options, and present conclusions.

Here is an example of math problem-based learning:

This is the four 4s math problem. The goal is to combine four 4s in various ways in order to make as many different integers as possible. The "combine" rules are that one can use addition, subtraction, multiplication, division, and parentheses.
Thus: (4 + 4)/(4+ 4) = 1; 4/4 + 4/4 = 2; and (4 + 4 + 4)/4 = 3.
For a more complex version of the problem, also allow concatenation (thus, 44/44 = 1 and 444/4 = 111), exponentiation, or other types of operations.

This math problem and its variations is widely used in math education. It illustrates that a math problem may have more than one solution. It illustrates the need for very careful definition of a problem. It is a problem that can engage individual students or a team of students over an extended period of time. Thus, you can see it has some of the characteristics of a project. However, typically all students are required to work on the exact same problem. Some students are likely to produce more answers than others.

There are many variations of the problem, both in the base number (for example, how about using four 3s) and the allowable operations. A Google search of four fours math problem produces many thousands of hits.

For more information about problem-based learning see:

Math Project-based and Problem-based Learning Is a Large Topic

PBL in math education is a large and challenging topic. A recent Google search of math "project-based learning" produced 90,100 hits. A Google search of math "problem-based learning" produced 86,400 hits. Many of these hits are documents covering both project-based and problem-based learning.

At the Northwest Mathematics Conference held October 9-11, 2008, in Portland, Oregon, each of the exhibitors was asked about the availability of project-based learning materials from their company. Many responded by pointing to certain sections in their textbook series. For the most part, the materials identified were better classified as problem-based learning rather than as project-based learning.

This little bit of evidence suggests that project-based learning is not a major topic in the current K-12 math curriculum.

Math Project-based and Problem-based Collaborative Learning

Both project-based learning and problem-based learning can be done by individuals or by teams. When done by teams, PBL is an example of collaborative learning. There has been substantial research on collaborative learning. Here are three short sections quoted from this collaborative learning reference:

Traditional academic approaches—those that employ narrow tasks to emphasize rote memorization or the application of simple procedures—won't develop learners who are critical thinkers or effective writers and speakers. Rather, students need to take part in complex, meaningful projects that require sustained engagement and collaboration.
Productive Collaboration. A great deal of work has been done to specify the kinds of tasks, accountability, and roles that help students collaborate well. In a summary of forty years of research on cooperative learning, Roger and David Johnson, at the University of Minnesota, identified five important elements of cooperation across multiple classroom models:
  • Positive interdependence
  • Individual accountability
  • Structures that promote face-to-face interaction
  • Social skills
  • Group processing
Evidence shows that inquiry-based, collaborative approaches benefit students in learning important twenty-first-century skills, such as the ability to work in teams, solve complex problems, and apply knowledge from one lesson to others. The research suggests that inquiry-based lessons and meaningful group work can be challenging to implement. They require changes in curriculum, instruction, and assessment practices—changes that are often new for teachers and students.

Project-based Learning Is a Team Activity

"Individual commitment to a group effort—that is what makes a team work, a company work, a society work, a civilization work." (Vince Lombardi; American football coach; 1913–1970.)

Project-based learning is a team learning and doing activity. Nowadays, a project-based learning team consists of:

  1. One or more students who are in charge of the project. They may be widely dispersed in location.
  2. People, such as peers, siblings, parents, teachers, and so on. They serve as advisers, sources of information, and sources of formative and summative feedback.
  3. Virtual and physical libraries. (The Web is a virtual library.)
  4. Tools, including computers, computer programs, and other Information and Communication Technology (ICT).
  5. Other resources such as materials, money, facilities, and environments in which a project is being carried out or is focused on.

At first glance, this may seem like a strange way to think about membership on a project team. The idea being emphasized is that project-based learning is always done in a team environment, even if there is only one student directly involved. Even a one-person team draws upon the accumulated knowledge and skills of a huge collection of people and other resources. The Web, for example, represents the past and continuing work of many millions of people. The tools (including computer tools) we routinely use represent the thinking and production work of a large number of people.

One of the most important goals in project-based learning is for students to learn to make effective use of these five different types of team members. It is a valuable life skill. Notice that this goal is independent of any specific content area that a project might focus on. The expertise one develops in working in this team environment readily transfers to other projects.

The strength or value of the fourth type of team member varies considerably from project to project, and from discipline to discipline. Sure, the Web is likely to be useful in almost any project in any discipline. But remember that math and computer science are strongly overlapping disciplines. Nowadays, calculators (4-function, scientific, graphic, equation-solving) and computers are an integral component of math education and of people making use of the math they are learning.

Another important goal is for students to gain increased confidence in their ability to accomplish complex tasks that are probably beyond their ability to accomplish without the aid of other "members" of the team. Such accomplishments help to build self-esteem.

Math is a Large and Old Discipline

"In most sciences one generation tears down what another has built and what one has established another undoes. In mathematics alone each generation adds a new story to the old structure." (Hermann Hankel; German mathematician; 1839-1873.)
"If I have seen further it is by standing on the shoulders of giants." (Isaac Newton; English mathematician & physicist; 1642-1727.)

The history of math predates the first development of reading and writing that occurred more than 5,000 years ago. The development of reading, writing, and math notational systems laid the foundations for the "basics" of reading, writing, and arithmetic in our current educational system. Thus, our formal educational system has more than 5,000 years of experience in developing and teaching math in schools.

Math is a broad and deep discipline that people learn through both informal and formal education. Each academic discipline or area of study can be defined by a combination of general characteristics such as:

  • The types of problems, tasks, and activities it addresses.
  • Its accumulated accomplishments such as results, achievements, products, performances, scope, power, uses, impact on the societies of the world, ability to attract followers and supporters, and so on.
  • Its history, culture, and language, including notation and special vocabulary.
  • Its methods of teaching, learning, assessment, and thinking. What it does to preserve and sustain its work and pass it on to future generations.
  • Its tools, methodologies, and types of evidence and arguments used in solving problems, accomplishing tasks, and recording and sharing accumulated results.
  • The knowledge and skills (the levels of math expertise; the levels of math maturity) that separate and distinguish among: a) a novice; b) a person who has a personally useful level of competence; c) a reasonably competent person, employable in the discipline; d) a regional or national expert; and e) a world-class expert.

A little thought should convince you that it is not easy to carefully define a particular discipline in a manner that differentiates it from other disciplines. Thus, it is appropriate to give students the following type of assignment:

Select a non-math discipline of study. Give a good definition of this discipline, and a good definition of the discipline of math. Compare and contrast the two disciplines. In this compare and contrast, make sure you cover all of the general defining characteristics of a discipline (see the bulleted list given above). Produce a report that will be presented to the whole class and that will be turned in to your teacher.

Relevance of the List to Math Project-based Learning

Math is a human endeavor. The current—and still rapidly growing—discipline of mathematics is the result of many tens of thousands of contributors over thousands of years.

Think about your math-related insights into the second item in the six-item list of defining characteristics of a discipline:

Its accumulated accomplishments such as results, achievements, products, performances, scope, power, uses, impact on the societies of the world, ability to attract followers and supporters, and so on.

What do you know about these aspects of math, and what do you want your students to know? These areas are not covered in state and national math assessments. However, they are important aspects of math as a human endeavor. They are a rich source of projects in a math course.

Think about your math-related insights into the third item in the list of defining characteristics of a discipline:

Its history, culture, and language, including notation and special vocabulary.

Math is often called a language. Learning to read, write, speak, listen, and communicate in this language is a critical aspect of learning math. Math journaling is a very useful and ongoing individual student project that contributes to students learning to communicate in the language of mathematics.

The history of calculators and computers is closely intertwined with the overall history of math. Calculators and computers were initially developed as aids to doing arithmetic. Now, of course, computers are important aids to representing and solving problems in every academic discipline. A math-centric way to think about this is that computers represent one of the ways that math is indispensable in every academic discipline. Certainly as one studies uses of math throughout the curriculum, calculators and computers must be given substantial emphasis.

The history and applications of math provide a wide range of possible projects for project-based learning.

Finally, consider the whole list of characteristics of a discipline, and think about them in terms of math and math education. When you are asked, "What is math?" what is your response? What response might your students give? What response would you like them to give? Clearly, the discipline of math is much more than the specific math topics you teach during math classes. Project-based learning can provide opportunities for students to greatly broaden their insights into the discipline of mathematics.

Goals of Math Education

“Mathematics consists of content and know-how. What is know-how in mathematics? The ability to solve problems.” (George Polya; Hungarian and American math researcher and educator; 1887–1985.)

Here's a common-sense suggestion to math teachers. Make use of math project-based learning to:

  • Help meet math learning goals that are better achieved in a project-based learning environment than through other ways to teaching math. Keep in mind that some of these goals are "traditional" while others may not be currently met in a traditional math curriculum.
  • Help students learn to do projects that involve math. This includes helping students learn some roles of math in the overall multidisciplinary area of project-based learning.

There are many possible goals in math education. Our math education system has identified some of these goals and emphasizes them in math curriculum content, instructional processes, and assessment. The next four sub-sections discuss math education goals.

Standards

In the United States, each state establishes its own goals (standards, benchmarks) for math education at the K-12 level. They are assisted in this endeavor by the work of the National Council of Teachers of Mathematics (NCTM) and a variety of other groups and organizations. The current Federal Government emphasis on math assessment is influencing state standards.

The individual states tend to establish a large number of math learning expectations at each grade level. For example, see the following table:

4th grade math LE.jpeg

Recent work of the NCTM has led to the development of a small number of Curriculum Focal Points for each PK-8 grade level. For example, the three Curriculum Focal Points for Grade 4 are:

  • Number and Operations and Algebra: Developing quick recall of multiplication facts and related division facts and fluency with whole number multiplication.
  • Number and Operations: Developing an understanding of decimals, including the connections between fractions and decimals
  • Measurement: Developing an understanding of area and determining the areas of two-dimensional shapes.

Both the individual states and the NCTM focus on learning specific math content. The curriculum being taught is strongly textbook driven. The textbooks provided by the leading publishers of math textbooks attempt to cover almost all of the topics listed by each state, and they also attempt to provide material that hits on a variety of the topics that help to define the discipline of mathematics. For example, in a typical textbook you are apt to find a number of short sections focusing on various mathematicians, including a picture of the mathematician and a little biographical information. Such textbooks also typically include numbers of short sections on applications of math in various occupations or non math disciplines.

George Polya

"A great discovery solves a great problem but there is a grain of discovery in the solution of any problem. Your problem may be modest; but if it brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery." (George Polya; Hungarian and American math researcher and educator; 1887–1985.)


There are many other ways to approach the question of goals for math education. For example, George Polya was a world class math researcher and math educator. In about 1969 he gave a talk on The Goals of Mathematics Education. Quoting from that talk:

Now what about mathematics teaching? Mathematics in the primary schools has a good and narrow aim and that is pretty clear in the primary schools. An adult who is completely illiterate is not employable in a modern society. Everybody should be able to read and write and do some arithmetic, and perhaps a little more. Therefore the good and narrow aim of the primary school is to teach the arithmetical skills—addition, subtraction, multiplication, division, and perhaps a little more, as well as to teach fractions, percentages, rates, and perhaps even a little more. Everybody should have an idea of how to measure lengths, areas, volumes. This is a good and narrow aim of the primary schools—to transmit this knowledge—and we shouldn't forget it.
However, we have a higher aim. We wish to develop all the resources of the growing child. And the part that mathematics plays is mostly about thinking. Mathematics is a good school of thinking. But what is thinking? The thinking that you can learn in mathematics is, for instance, to handle abstractions. Mathematics is about numbers. Numbers are an abstraction. When we solve a practical problem, then from this practical problem we must first make an abstract problem. Mathematics applies directly to abstractions. Some mathematics should enable a child at least to handle abstractions, to handle abstract structures. Structure is a fashionable word now. It is not a bad word. I am quite for it.
But I think there is one point which is even more important. Mathematics, you see, is not a spectator sport. To understand mathematics means to be able to do mathematics. And what does it mean doing mathematics? In the first place it means to be able to solve mathematical problems. For the higher aims about which I am now talking are some general tactics of problems -- to have the right attitude for problems and to be able to attack all kinds of problems, not only very simple problems, which can be solved with the skills of the primary school, but more complicated problems of engineering, physics and so on, which will be further developed in the high school. But the foundations should be started in the primary school. And so I think an essential point in the primary school is to introduce the children to the tactics of problem solving. Not to solve this or that kind of problem, not to make just long divisions or some such thing, but to develop a general attitude for the solution of problems.

You can see considerable overlap between Polya's ideas and those of individual states and the NCTM. But Polya's ideas tend to move beyond those of the individual states and the NCTM. He emphasized learning to think mathematically, to deal with math abstractions, and to deal with complicated problems. This type of math education is designed to help develop a student's level of math maturity.

Math Maturity

"A growing level of math maturity is shown by 'a significant shift from learning by memorization to learning through understanding.'" (See Wikipedia attribution below.)

Quoting from the Wikipedia: "Mathematical maturity is a loose term used by mathematicians that refers to a mixture of mathematical experience and insight that cannot be directly taught, but instead comes from repeated exposure to complex mathematical concepts." Still quoting from the Wikipedia, other aspects of mathematical maturity include:

  • the capacity to generalize from a specific example to broad concept
  • the capacity to handle increasingly abstract ideas
  • the ability to communicate mathematically by learning standard notation and acceptable style
  • a significant shift from learning by memorization to learning through understanding
  • the capacity to separate the key ideas from the less significant
  • the ability to link a geometrical representation with an analytic representation
  • the ability to translate verbal problems into mathematical problems
  • the ability to recognize a valid proof and detect 'sloppy' thinking
  • the ability to recognize mathematical patterns
  • the ability to move back and forth between the geometrical (graph) and the analytical (equation)
  • improving mathematical intuition by abandoning naive assumptions and developing a more critical attitude

Quoting Larry Denenberg:

Thirty percent of mathematical maturity is fearlessness in the face of symbols: the ability to read and understand notation, to introduce clear and useful notation when appropriate (and not otherwise!), and a general facility of expression in the terse—but crisp and exact language that mathematicians use to communicate ideas. Mathematics, like English, relies on a common understanding of definitions and meanings. But in mathematics definitions and meanings are much more often attached to symbols, not to words, although words are used as well. Furthermore, the definitions are much more precise and unambiguous, and are not nearly as susceptible to modification through usage. You will never see a mathematical discussion without the use of notation!

One of the most important goals in math education is to help students gain in their levels of math maturity. However, math maturity is difficult to assess, and thus is seldom assessed.

Enjoying Math

"We cannot hope that many children will learn mathematics unless we find a way to share our enjoyment and show them its beauty as well as its utility." (Mary Beth Ruskai; American mathematician and researcher; 1944-.)

Many adults have had math education experiences that lead them to say things like, "I hate (or, hated) math" and "I can't do math." Many people believe that a math education that produces the "I hate math" and related types of results is significantly flawed.

Thus, math teachers might want to place greater emphasis on helping students to enjoy and appreciate the math they are learning and the math learning experiences.

"The point is to make math intrinsically interesting to children. We should not have to sell mathematics by pointing to its usefulness in other subject areas, which, of course, is real. Love for math will not come about by trying to convince a child that it happens to be a handy tool for life; it grows when a good teacher can draw out a child's curiosity about how numbers and mathematical principles work. The very high percentage of adults who are unashamed to say that they are bad with math is a good indication of how maligned the subject is and how very little we were taught in school about the enchantment of numbers." Alfred Posamentier, Professor of Mathematics Education at the City College of New York, 2002 New York Times article.

Folk Math

Math is taught in schools throughout the world. However, there are many children who never attend school. Many others receive only a few years of formal education. A number of people have done studies of street-people who have learned the math they need without benefit of much (or, perhaps any) formal schooling.

Gene Maier has written excellent materials on the idea of Folk Math. Think in terms of folk music or folk art—knowledge and skills that are learned in informal settings but that meet the needs of the learners. Also, think in terms of the math that children learn as they memorize counting rhymes, play board games, card games, and computer games even before they start school.

In terms of project-based learning, think about designing a project for students to explore two areas:

  • What math do people learn informally (on the streets) in various places throughout the world? Pay special attention to children who receive little or no formal schooling.
  • What math do students learn in school in different parts of the world? How is this the same and how is it different from the math that students are learning in the United States?

The topic of Folk Math raises other issues. What math do typical adults with a high school education or more use in their everyday lives? For example, do their everyday demands of work, play, and other activities require quite a bit of high school algebra and geometry? This question can be the basis of project-based learning in which students interview adults and individually or jointly compile the results.

Another related topic is the question of how calculators, "cash" registers, and other devices with built-in computational capabilities have changed the math education needs of adults. In a typical day, what math does a typical adult actually do mentally, using pencil and paper, or using calculator/computer aids? Again, this is a project that can be done by individual students, but is quite well suited to teams of any size.

Still another type of project related to folk math is to explore how people did math thousands of years ago and in different parts of the world. How did merchants and customers do the math they needed to do when they did not know how to read and write? The abacus is a piece of this history.

"Why Learn This?" Projects

As students progress upward through the elementary, middle, and high school math curriculum, many eventually ask questions similar to, "Why am I being expected to learn this?" and "What's in it for me?"

These types of questions can be the basis for ongoing, individual projects. There are two general types of answers. First, there are the assertions that, "It will be good for you." and "You will need it in the future." Second, there are the personal answers that a student can develop for him or her own self and that are specifically tailored to the student's personal needs and ambitions.

A teacher can provide the "assertion" answers, and may be able to provide some general-purpose answers that tie in with the interests of some students. An example of such an answer is, "If you want to be a (teacher names several careers) when you grow up, you will need to take and pass math courses that are based on this material." Thus, for example, people majoring in engineering in college have to take calculus and other "higher" math.

The idea underlying making this into a math project-based learning assignment (possibly a project that continues all year) is to put the responsibility onto the student. Encourage the students to think about current and possible future mathematical needs each may have. Get them to analyze the math curriculum from a personal future point of view. The student selects a possible vocation to be pursued after leaving school. The project is to explore possible math-related needs in this vocational area and work to achieve a level of expertise that fits their personal needs and/or the needs of possible employers.

The goal is to get students to explore the vocation in some depth. What does the possible job or career contribute to people who work in the area, and what does it contribute to the world? Why is it important, and how is it self-satisfying and fulfilling?

Projects for Use in Math Education for Teachers

"The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it and he delights in it because it is beautiful." (Henri Poincare; French physicist and mathematician; 1854–1912.)

This section is especially aimed at people who teach preservice and inservice courses and workshops for math teachers. The theme is one of role modeling the use of project-based learning in a manner that will help students make immediate use of the information and material in their own learning and in the future as they work with the students they teach.

A number of topic areas are listed below. Each can be the basis for individual or small group projects designed to produce products and representations that can be shared with current fellow students and made available to other current and future preservice and inservice teachers.

  • Math and/or math education humor.
  • Math and/or math education history. If a student is going to learn one important "tidbit" of math history each year (or, half-year, or quarter-year) in our math education system, what should these tidbits be. Why?
  • Math and/or math education quotes. Ask and answer the same underlying question as for the line just above.
  • Math education research or general education research that is "powerful" enough to lead to significant improvements in math education.
  • Women and/or any other underrepresented population in math.
  • Math modeling and computational thinking. What are these two topics, and to what extent do they overlap? Develop some illustrations of use of computational thinking in learning the math you are currently studying or have studied in the past.
  • Math maturity, knowledge, and skills self-assessment.
  • Roles of calculators and computers in math and/or math education.

A key idea is to distinguish between a strongly problem-based learning orientation in math education and the traditional curriculum. For most students, a problem is a modestly challenging activity that is completed in a few seconds or perhaps a few minutes. Contrast this with the types of problems one encounters in upper-division undergraduate math courses and in graduate school math courses!

Ideas from Bob Albrecht

Project-based math has been Bob's passion since 1962, probably because he has a B.S. in physics and an M.S. in applied math with a minor in physics. Learn more about Bob Albrecht at:

The following is from Albrecht's 9/26/08 email to Moursund:

C-TEC was a project-based learning community within a high school, 200+ students in a school of 1600 students. We spent 6 years as a project mentor at C-TEC. Some students called us Bob; others called us George.
Read our Starship Gaia materials published in Learning and Leading with Technology and now available at Curriki. Go to www.curriki.org and search for starship gaia.
Browse Investigation Backpack at Curriki. Go to www.curriki.org and search for investigation backpack.
Browse Solar System at Curriki. Go to www.curriki.org and search for albrecht solar system.
Solar System—Inner in Learning and Leading with Technology, 2005-2006.
Starship Gaia: Model the Motions of the Planets in Learning and Leading with Technology, 2008.

References and Resources

Blumenfeld, P., Soloway, E., Marx, R., Krajcik, J., Guzdial, M., & Palincsar, A. (1991) Motivating project-based learning: Sustaining the doing, supporting the learning. Educational Psychologist, 26 (3 & 4).

Ewen, L. (n.d.). Summary of Blumfeld, et al., article. Retrieved 10/8/08: http://mathforum.org/~sarah/Discussion.Sessions/Blumenfeld.html. Quoting from the article:

There are two essential components of projects: They require a question or problem that serves to organize and drive activities; and these activities result in a series of artifacts, or products, that culminate in a final product that addresses the driving question.
The authors stress that giving students freedom to generate artifacts is critical to their construction of knowledge. Whether the guiding questions and activities are student- or teacher-generated, their outcomes must not be fixed at the outset or students will not have the opportunity to try their own problem-solving approaches.

Brown, E., & Harrington, T. (2003). Project Based Learning: Mathematics in the Real World. In Lassner, D., & McNaught, C. (eds.), Proceedings of World Conference on Educational Multimedia, Hypermedia and Telecommunications 2003. Chesapeake, VA: AACE. Retrieved 10/6/08: http://www.editlib.org/INDEX.CFM?fuseaction=Reader.ViewAbstract&paper_id=11134.

Abstract. This paper describes an overview of an exemplary unit developed during an Intel Teach to the future Faculty Pre-Service Training utilizing a constructivist's project based learning approach (PBL) to mathematics in the real world as espoused by state and national standards. The PBL unit creation is a highly motivating, multi-sensory project spanning the realm of multiple intelligences to actually make classroom mathematics "come alive" while demonstrating various aspects based on an essential question—"Where is the Mathematics in the Real-World." Scaffolding in the form of unit questions enhanced by technology integration directly linked to the essential question provides a fascinating unit on mathematics in the real world.

Miller, A. (6/28/2011). Assessing the Common Core Standards: Real life mathematics. educopia. Retrieved 6/29/2011 from http://www.edutopia.org/blog/assessing-common-core-standards-real-life-mathematics. This paper includes three useful links:

NMSA (n.d.). Project-based learning in middle grades mathematics. National Middle School Association. Retrieved 10/6/08: http://www.nmsa.org/portals/0/pdf/research/Research_Summaries/ProjectBased_Math.pdf. Quoting from the document:

The focus of this research summary is to foster an understanding of project-based learning (PBL), particularly in mathematics education; to explain the factors for making a conscious decision to implement PBL in middle grades mathematics classrooms; and to provide insights about the possible realized effects when mathematics-based PBL is implemented.
The teacher’s belief system is paramount. A teacher who believes that social constructivism (Vygotsky, 1978) or situated learning (Boaler, 1999; Cobb, 1988) is useless, will find the work and effort for accomplishing PBL to outweigh its benefits. The tenets of constructivism, in its many versions, underlie PBL designs (Grant, 2002). In PBL, the teachers’ role necessitates that they allow all students to engage in developing personally and collaboratively negotiated meanings from the learning event (Harel & Papert, 1991; Kafai & Resnick, 1996). The success of PBL should be assessed on many levels including emotional development, collaboration, leadership, and negotiation skills that are essential for project success (Glaser, 1992; Glennan, 1998). When teachers allow students more autonomy over what they learn, it improves motivation, and students assume more responsibility for their learning (Tassinari, 1996; Wolk, 1994; Worthy, 2000). However, this does not mitigate the important need for the teacher to be actively engaged in the learning task as both a role model and an advisor. This role also does not forsake whole-group didactic instruction, but makes careful use of it to address learning deficits. The teacher can function as a co-constructor of knowledge (Rosenfeld & Rosenfeld, 2006). In this role, the teacher must possess profound content knowledge, be confident in his or her skill to facilitate learners of diverse abilities, and be prepared to deal with a more diverse set of questions—potentially across disciplines. Consequently, the role of the teacher and this diversity in content raise questions about the scope and style of assessing student learning.

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Author or Authors

The original version of this document was developed by David Moursund. A number of people—and especially Bob Albrecht—have suggested ideas and materials for inclusion in this document.

Ideas Not Yet Incorporated in the Document

Here are some quotations that might be useful:

“Mathematics is the queen of the sciences.” (Carl Friedrich Gauss; German mathematician, physicist, and prodigy; 1777–1855.)
"At the age of eleven, I began Euclid, with my brother as my tutor. This was one of the great events of my life, as dazzling as first love. I had not imagined there was anything so delicious in the world." (Bertrand Russell; English philosopher, historian, logician, and mathematician; 1872–1970.)
“When you are up to your neck in alligators, it's hard to remember the original objective was to drain the swamp." (Adage, unattributed.)

Here are some key ideas that could be incorporated into this document:

1. Teaching for high-road transfer of learning. As students learn a strategy for use in solving math problems, they need to search for applications in the other disciplines they know. This could be a whole class project, with materials added each time a new math problem-solving strategy is explored. The product might be a website or a short book of lasting value that can be shared and later expanded on from year to year.

2. The general idea of projects that are "fully" math versus projects that are in other disciplines, but are ones that use a lot of math.

3. Uses of math in addressing "big" problems such as sustainability. A great many people agree that sustainability is a major problem facing the people of our world. Quoting from the Wikipedia:

One of the first and most oft-cited definitions of sustainability, and almost certainly the one that will survive for posterity, is the one created by the Brundtland Commission, led by the former Norwegian Prime Minister Gro Harlem Brundtland. The Commission defined sustainable development as development that "meets the needs of the present without compromising the ability of future generations to meet their own needs." The Brundtland definition thus implicitly argues for the rights of future generations to raw materials and vital ecosystem services to be taken into account in decision making. [Bold added for emphasis.]

In some sense, the life of a typical K-12 teacher (of math and/or other subjects) is quite a bit like being up to one's neck in alligators. It is easy to lose track of still larger challenges such as global problems of sustainability, large numbers of people living in poverty, and large numbers of people having totally inadequate medical care.

One of the over-arching goals of education is to help students grow up to be responsible adults who can contribute to helping to solve such problems. Thus, each teacher has some responsibility to help students gain knowledge, skills, and maturity as they move toward becoming responsible adults.

With rare exceptions, teachers tend to be good role models. Their daily interactions with students help the students to move toward becoming responsible adults.

However, there is much more that teachers can be doing. Let's use the problem of sustainability as an example. What can a math teacher do to help students learn about the problem of sustainability and gain the math-related knowledge, skills, and maturity to help address this problem?

Carbon Footprint. The basic ideas include:

1. Helping students to understand the problem and its importance.

2. Helping students to understand the need for measurements, forecasts, and problem solving in attempting to deal with the problem.

3. Helping students to understand what they individually and collectively, at school, at home, and in the community, can do to help address the problem.

4. Helping students to understand the value and power of teamwork, as they work together as teams of groups of students, the whole class, the whole school, and so on over a long period of time.

5. Helping students learn more about the world and how sustainability is a worldwide problem.

Students can learn about what others are doing. See, for example, http://www.guardian.co.uk/education/2008/jul/16/schools.uk2 and http://education.guardian.co.uk/egweekly/story/0,,2006238,00.html

Conspicuous Consumption. Many math textbooks used at the precollege level include examples that can be construed as being related to sustainability. For example, here is a word problem:

Mary wants to buy a cashmere sweater that costs $62. She has an allowance of $3.50 a week. She figures that she can save $2 per week from this allowance. How many weeks will it take her to save enough money to buy the sweater?

Notice that this is a consumption-oriented problem situation. Mary has income and she "wants" to buy a cashmere sweater. There is no indication of why she wants the sweater or if she "needs" the sweater. What is cashmere? Are there other types of sweater fabric that are more supportive of sustainability? How does this tie in with Mary's carbon footprint?

Aha! At what age can students begin to understand some of the basic ideas of sustainability, carbon footprint, ways to save energy, and so on? Are there math problems that are appropriate to the math curriculum for young students and that can help to teach and support sustainability?

It seems to me that there are two goals here:

  1. Help preservice and inservice elementary school teachers of math realize that they can and should be teaching students about math-related aspects of sustainability.
  2. Provide some sources for materials and resources that will help teachers to implement such ideas.

There are a number of carbon footprint calculation websites. I don't know how much math education material based on sustainability ideas and keyed to the math curriculum is available.